Design Platform for Planar Mechanisms Based on a Qualitative Kinematics
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Design Platform for Planar Mechanisms based on a Qualitative Kinematics Bernard Yannou Adrian Vasiliu Laboratoire Productique Logistique Ecole Centrale Paris 92295 Chatenay-Malabry, France phone +33 141131285 fax +33 141131272 email [yannou I adrian]@cti.ecp.fr Abstract: Our long term aim is to address the open problem of combined structural/dimensional synthesis for planar mechanisms. For that purpose, we have developed a design platform for linkages with lower pairs which is based on a qualitative kinematics. Our qualitative kinematics is based on the notion of Instantaneous Centers of Velocity, on considerations of curvature and on a modular decomposition of the mechanism in Assur groups . A qualitative kinematic simulation directly generates qualitative trajectories in the form of a list of circular arcs )out and line segments. A qualitative simulation amounts to a n slider-crank propagation of circular arcs along the mechanism from the Figure 1 : Example of a linkage, composed of a mechanism (bodies 1, 2 and 3) and a dyad (bodies 4 and 5). Joints A motor-crank to an effector point. A general understanding and G are revolute joints (rotation) connected to the fixed frame. of the mechanism, independent of any particular Joint D is a prismatic joint connected to the fixed frame. The dimensions, is extracted . This is essential to tackle other joints are revolute joints connecting two bodies. There is a synthesis problems . Moreover, we show that a motor atjoint A, so body 1 is a crank dimensional optimization of the mechanism based on our qualitative kinematics successfully competes, for the path a requirement of desired kinematics in the form of the desired trajectory effector point, generation problem, with an optimization based on a of a point, termed on conventional simulation. a body of the mechanism, making abstraction of the orientation of this body. This problem is called the path generation problem. 1 Introduction A synthesis process consists of choosing the type of Mechanical linkages with lower pairs are planar mechanism (type synthesis), e.g., slider-crank or four-bar mechanisms of fixed topology, consisting of rigid bodies mechanism, and the optimal dimensions for this mechanism constantly connected with joints of revolute type (allowing (dimensional synthesis). The dimensional synthesis a rotation) or prismatic type (allowing a translation). An problem has been widely studied for two decades and more example is given in Fig. 1 . Linkages with lower pairs [13; 25], and some satisfactory techniques have been represent a large part of the existing mechanisms in implemented. However, the combined type/dimension industry. Our subject is to develop an efficient automatic or synthesis problem remains an open problem for two major semi-automatic method for the synthesis of these linkages. reasons. On one hand, it often consists in testing a huge Such a method should find one or several optimal number of simple equivalent mechanisms in terms of a mechanisms submitted to functional requirements. These unique criterion: the number of degrees of freedom [12; 24; requirements are conventionally: 26] . On the other hand, the criteria which preside over the pruning of such a combinatory are completely " the desired kinematics of one or several bodies, questionable. In fact, the choice of the type of mechanism is " no collision between the mechanism and its very little constrained by the requirements and, for a given environment, type of mechanism, there is no classification of the trajectories in terms of the dimensions. This reveals that at " the minim;zation of the efforts in joints when the two previous criteria are fulfilled. present we have very poor and unstructured knowledge about the interactions between functions, structures and For the moment, we have limited our research to: dimensions for the mechanisms. " linkages of one-degree-of-freedom, It is for this reason that we developed a platform for the design of linkages which is based on a qualitative kinematics. The idea of this qualitative kinematics is to In the case of the generation of qualitative explanations, consciously accept a small loss in numerical accuracy of the exact trajectory is generated by a primaryconventional trajectories, in order to handle more global and qualitative kinematic simulation for specific dimensions of the information and to have a deeper understanding of mechanism which must be explained . This is the trajectory mechanisms . This is made by directly generating qualitative of the effector point which intervenes in the specification, trajectories in the form of lists of circular arcs and line i.e., in the function of the mechanism, so that we can say segments, instead of generating lists of points like in a that the sought function drives the explanations . The conventional simulation . process of the generation of the qualitative explanations is Firstly, we show that a dimensional optimization summed up in Fig. 2- (synthesis) process based on (looping over) such a qualitative kinematic simulation is faster than an optimization process based on a conventional simulation. For brevity, we will term these two dimensional optimization processes : qualitative optimization and exact (or conventional) optimization . Secondly, and this is the real challenge, we show how to construct a qualitative understanding of a mechanism as an engineer could do, in order to justify the qualitative shape of the trajectory from particular dimensions . Only the work of Shrobe [27] deals with this purpose but in a less explicit manner . The qualitative explanations that our system generates use two Figure 2: Generating qualitative explanations of the behaviour of a mechanism . mechanisms of causal deductions: a modular decomposition of the mechanism and a causal graph of Instantaneous In the case of qualitative optimization (which is a loop Center of Velocityl calculations. Thirdly, we hope that, in over a qualitative kinematic simulation), the exact the long term, we shall be able to invert the explanation trajectory is specified by the designer himself because this process . Then, we would be able, in a combined is the desired trajectory to fulfill . After a qualitative type/dimension synthesis process, to intelligently index dimensional optimization, optimal dimensions for the some mechanism types which are serious candidates and to mechanism are proposed : this is the qualitative optimum directly propose value intervals for the dimensions . (see Fig. 3) . In order to confirm the validity ofour approach Section 2 presents the two functionalities of the design (in spite of the two consecutive approximations : first platform : generation of the qualitative explanations and approximation into arcs and segments and qualitative qualitative optimization . We evoke, in section 3, the simulation), we will have to compare the qualitative modular decomposition of a mechanism and its advantages . optimum with the exact optimum directly obtained by a In section 4, we present our qualitative kinematics. Section conventional optimization . 5 shows how our qualitative kinematics can be applied to a dimensional optimization . The most important section is 'Mechanism type' section 6 where we show how to construct qualitative WITHOUT any explanations for a mechanism. Finally, in section 7 we specific dimension compare our work with others on qualitative kinematics. Approximation 2 The two functionalities of the design EXACT into arcs and QUALITATIVE desired traj. platform segments d6sired traj . 2.1 Qualitative explanations and qualitative dimensional optimization Both functionalities of the design platform (generation of VALIDATION QUALITATIVE optimum the qualitative explanations and qualitative optimization) require as a first step an approximation of an exact Figure 3: Qualitative dimensional optimization. The method trajectory (in the form of a list of points) into a qualitative remains to be validated by comparison between the qualitative trajectory (in the form of a list of arcs and segments) . In and the exact optima . both cases, this resulting qualitative trajectory is used to drive a qualitative kinematic simulation. 2.2 Approximation of an exact trajectory into arcs and segments This approximation takes as input argument a maximum 1 Henceforward, they are termed ICs. admissible error. This error is the maximum distance Figure 4: Approximation of an exact trajectory (left) defined by 48 points into a qualitative trajectory (right) defined by 4 circular arcs. between a point of the exact trajectory and the qualitative trajectory. A minimization of the total number of arcs and segments is carried out with a least square algorithm . For the same total number of arcs and segments, the maximum number of segments is preferred because they have more Figure 6: A systemic representation of the linkage of Fig. 1 by a significant qualitative properties . Finally, the third modular decomposition into multipole groups. criterion is to prefer the approximation which minimizes considered as a sub-system, receives from the previous the error. The algorithm does not respect the continuity group information about the position, velocity and between the arcs (and segments) because it is not necessary. acceleration of its input joints. After an internal An example of approximation is given in Fig. 4. calculation according to the transfer function of the group, the kinematic features of the output poles are determined 3 Modular decomposition of linkages and transmitted to